Restricted numerical range: a versatile tool in the theory of quantum information

TL;DR

This paper introduces a modified numerical range concept for Hermitian operators, considering specific subsets of quantum states, and demonstrates its applications in quantum information theory, including entanglement and channel analysis.

Contribution

It develops the concept of restricted numerical range for various quantum state subsets and applies it to problems like entanglement detection and local distinguishability of unitary gates.

Findings

Analyzed k-positive maps and entanglement witnesses using restricted numerical ranges.

Applied product numerical range to solve local distinguishability of unitary gates.

Provided insights into minimal output entropy of quantum channels.

Abstract

Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers for instance the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.